Atomic potential energy uncertainty in machine-learning interatomic potentials and thermal transport in solids with atomic diffusion

Thermal transport simulations have attracted wide attention in recent years, and one standard approach is to use the Green-Kubo method based on machine-learning interatomic potentials and equilibrium molecular dynamics (GK-MLIP-EMD). In this work, we focus on the lattice thermal conductivities ${\ensuremath{\kappa}}_{L}\mathrm{s}$ for solids with atomic diffusion by taking $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{C}{\mathrm{u}}_{2\text{\ensuremath{-}}x}\mathrm{Se}$ ($0\ensuremath{\le}x\ensuremath{\le}0.05$) as an example. Surprisingly, the GK-MLIP-EMD approach fails in the evaluation of ${\ensuremath{\kappa}}_{L}\mathrm{s}$ for $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{C}{\mathrm{u}}_{1.95}\mathrm{Se}$, whereas the direct method based on nonequilibrium molecular dynamics reliably predicts these values instead. The failure of GK-MLIP-EMD for $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{C}{\mathrm{u}}_{1.95}\mathrm{Se}$ could be attributed to the ambiguous projection of the local atomic potential energy ${U}_{i}$ in MLIPs, exacerbated by the Cu diffusion at elevated temperatures. The Cu diffusion in $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{C}{\mathrm{u}}_{1.95}\mathrm{Se}$ greatly increases the ratio of the convective term and the uncertainty of the conductive term. These influences are considered negligible in crystalline solids. Our findings imply that the ambiguous definition of ${U}_{i}$ in MLIPs breaks down the applicability of the GK-MLIP-EMD approach to ${\ensuremath{\kappa}}_{L}$ prediction for solids with severe atomic diffusion.